Topological formulation of homology and cohomology theories

Exercise 3.1.5. Show that precomposition with \(\Pi _{\infty }\colon \Top _* \to \An _*\) establishes a bijection between reduced cohomology theories in the sense of Definition 3.1.2 and pairs \((E^*,\sigma )\) consisting of a functor \(E^*\colon \Top _*\catop \to \Ab ^{\Z }\) and a natural isomorphism \(\sigma \colon E^*(-) \iso E^{*+1}(\Sigma (-))\) satisfying:

(1)

Weak homotopy equivalences induce isomorphisms on \(E^*\).

(2)

The wedge axiom holds for wedges of pointed CW-complexes whose basepoints are \(0\)-cells.

(3)

For every relative CW-pair \((X,A)\), the sequence \(E^*(X/A) \to E^*(X) \to E^*(A)\) is exact.

Formulate the analogous characterization of homology theories.

Hint: Use Theorem 2.3.4 to reduce to CW-complexes, and Proposition 2.4.12 to transport the relevant constructions.